Question: Alexis wants to make a paperweight at pottery class. He designs a pyramid-like model with a base area of $100$ square centimeters and a height of $6$ centimeters. He wants the paperweight to weigh at least $300$ grams. What is the lowest possible density of the material Alexis uses to make the paperweight?
Solution: This is a density word problem. To solve it, we can use the following equation, which is the volume definition of density: ${\text{Density}}=\dfrac{{\text{Total quantity}}}{{\text{Volume}}}$ What do we know? The paperweight's weight should be at least ${300}$ grams, which is the ${\text{total quantity}}$. The base area of the pyramid-like paperweight is $100$ square centimeters and its height is $6$ centimeters (we can use this to find the ${\text{volume}}$ ). What do we need to find? The lowest possible ${\text{density}}$ of the material Alexis uses to make the paperweight. We want the ${\text{total quantity}}$ to be at least ${300}$ grams. This means we need to rewrite the equation so the ${\text{total quantity}}$ is isolated: $\begin{aligned} {\text{Density}}&=\dfrac{{\text{Total quantity}}}{{\text{Volume}}} \\\\ {\text{Density}}\cdot{\text{Volume}}&=\dfrac{{\text{Total quantity}}}{\cancel{{\text{Volume}}}}\cdot\cancel{{\text{Volume}}} \\\\ {\text{Density}}\cdot{\text{Volume}}&={\text{Total quantity}} \end{aligned}$ So this is the inequality we want to solve: ${\text{Density}}\cdot{\text{Volume}}\geq{300}$ The ${\text{volume}}$ of the pyramid-like paperweight is $\dfrac{1}{3}\cdot 100\cdot 6={200}$ cubic centimeters. Let's denote the density as $ d$. Now we can plug ${\text{density}=d}$ and ${\text{volume}=200}$ in the inequality. $\begin{aligned} {\text{Density}}\cdot{\text{Volume}}&\geq{300} \\\\ {d}\cdot{200}&\geq{300} \\\\ d&\geq\dfrac{{300}}{{200}} \\\\ d&\geq 1.5 \end{aligned}$ The lowest possible density of the material Alexis uses to make the paperweight is $1.5$ grams per cubic centimeter.